Ptolemy (85 – 165), Mathematician, Geographer, Astronomer and Astrologer

Cellarius ptolemy system

Claudius Ptolemy, Mathematician, Geographer, Astronomer and Astrologer

Born in Ptolemais (Upper Egypt), hence his name, this illustrious astronomer and geographer lived in Alexandria. His masterful work, the Mathematical Composition (around the year 150), thirteen books, renamed Megiste Syntaxis (the very large treatise), and the Almagest by Arab astronomers and mathematicians (from al = le and megistos = very large), is an extensive compilation of the hypotheses and results obtained at the time on the motion of celestial objects, supplemented by a treatise on plane and spherical trigonometry based on string theory. In his impressive treatise on Geography, we owe him the notions of latitude and longitude of a terrestrial place.


Mathematician, philosopher, musicologist, epigrammatist, music theorist, astronomer, astrologer, geographer, writer.

All his discoveries

Ptolemy is the author of two major works which summarize the astronomical and geographical knowledge of his time. He is also the author of a theorem that bears his name.

Ptolemy cataloged many stars, assigning them brightness and magnitude, and established rules for predicting eclipses.

Ptolemy’s theorem

Ptolemy’s theorem: For a cyclic quadrilateral (that is, a quadrilateral inscribed in a circle), the product of the diagonals equals the sum of the products of the opposite sides. AC BD = AB CD + AD BC.

Ptolemy equality
Ptolemy’s theorem is a relation among these lengths in a cyclic quadrilateral. Kmhkmh, CC BY 3.0, via Wikimedia Commons

Ptolemy’s Astrological Zodiac

The traditional Zodiac of the constellations is not a firm enough base, in the eyes of Ptolemy, to establish his method in astrology in the Tetrabible, because it is subject to the precession of the equinoxes*, which leads to a slow displacement of the four cardinal points of the Zodiac, namely the points of the equinoxes and the solstices. Thus, the vernal equinox which occurred in the constellation of Taurus, two thousand years BC. AD, then in the 1st century AD. BC in the constellation of Aries, takes place today in the constellation of Pisces. To avoid this continuous discrepancy, Ptolemy detaches, from this Zodiac of constellations, a fictitious Zodiac which he attaches forever to the point of the spring equinox and thereby to the cycle of the seasons. Also each year at the time of spring (March 21), it will always be, according to this conception, the sign of Aries in this fictitious Zodiac.

Hipparchus and the precession of the equinoxes

Hipparchus, the famous astronomer, discovered in the 2nd century BC. AD, the phenomenon of the precession of the equinoxes. This will only find its full explanation much later with Newton and the concept of universal attraction.

This precession of the equinoxes is caused by disturbances due to the joint attraction of the Moon and the Sun on the terrestrial globe. Thus, the axis of rotation of the globe does not keep the same direction, but describes very slowly, within a period of 26,000 years, a cone whose half-angle at the top is approximately 24°.

Also, the celestial pole is not fixed on the celestial sphere, and the equinoctial points shift on the ecliptic by 50° per year from East to West. Thus in their annual meeting the equinoctial point is ahead of the Sun and “precedes” it (hence the term precession) by twenty minutes each year.

The scheme of the aforementioned division of spheres. · The empyrean (fiery) heaven, dwelling of God and of all the selected · 10 Tenth heaven, first cause · 9 Ninth heaven, crystalline · 8 Eighth heaven of the firmament · 7 Heaven of Saturn · 6 Jupiter · 5 Mars · 4 Sun · 3 Venus · 2 Mercury · 1 Moon. Fastfission, Public domain, via Wikimedia Commons


Ptolemy also wrote the Harmonics, a landmark musicological treatise on the theory and mathematical principles of music.

After a criticism of the approaches of his predecessors, Ptolemy pleads there to base musical intervals on mathematical proportions (contrary to the partisans of Aristoxène) supported by empirical observation (contrary to the purely theoretical approach of the Pythagorean School). He presented his own divisions of the tetrachord and the octave, which he derived with the help of a monochord. Ptolemy’s interest in the theory and principles of harmony in music also appears in a discussion of the music of the spheres.

The treatise on Harmonics will contribute to the development of the musical theory of Boethius De Institutione Musica in the sixth century.

Discover his biography

Claudius Ptolemy was born around the year 90 AD in Upper Egypt, he died around the year 168. Although very little is known about his life, his name has gone down in history because of his two major contributions, a work on astronomy, the Almagest, and another entitled Geography.

In the Almagest, Ptolemy brings together nearly nine centuries of astronomical observations accumulated by the Babylonians and the Greeks: his tables of positions will serve as a reference until the appearance of the astronomical telescope in the 17th century. Ptolemy supports the geocentric model of the Universe proposed by Hipparque, a model which puts the Earth at the center of everything and which the Church will only abandon in the 18th century. With a catalog of more than 1000 stars and a list of 48 constellations, the Almagest is the only work of astronomy from antiquity that has come down to us, leaving a lasting mark on Western and Eastern science.

In his Geography, Ptolemy realizes for the Earth what he did for the Sky in the Almagest: a compilation of the geographical knowledge of the world, based on the records of the Roman geographer Marinus of Tyr. To facilitate the unification of the maps in his possession, Ptolemy imagines a system of coordinates which always makes reference, by measuring the latitudes from the equator and the longitudes by fixing the meridian 0 in the Fortunata islands (probably the islands of Green cap).

Finally Ptolemy is also known for his mathematical work and his theorem (in a convex quadrilateral inscribed in a circle, the product of the diagonals is equal to the sum of the products of the opposite sides) as well as for his studies on the properties of light (color , reflection, refraction and theory of vision).

The Almagest: the major work of Ptolemy

Ptolemy was part of the Aristotelian school and for this reason, he placed the earth at the center of the universe.

Inspired by the work of Hipparchus, the Almagest (Book by Ptolemy) describes harmonious geocentrism to Aristotle: Earth, motionless, is the center of the world rotating in a circle and at a uniform speed, the Moon then Mercury, Venus, the Sun, Mars (whose orbital eccentricity is reminiscent of Kepler’s). 1500 years later on the merits of heliocentrism), Jupiter and Saturn and finally, far beyond, the stars. In the Almagest Ptolemy presented descriptions of the 48 classic constellations and created an enhanced system to explain the apparent motion of the planets in a geocentric system in which the Sun, Moon and planets revolve around the Earth in circles called epicycles. The Almagest consists of 13 volumes.

His major work, which he had previously named “mathematical composition or syntax” but which was retranslated by Arabic translators, is the Almagest (Al in Arabic followed by a Greek superlative meaning “the greatest”). This book sets out a Ptolemaic theory of the organization of the world, a work that has been copied and translated many times into Arabic, Latin, Greek(…) and distributed throughout the Mediterranean basin, whose immense influence will last 1,500 years.
This huge treatise is made up of thirteen books divided into very distinct themes.

Almagest 1
Figure from a translation of the Almagest in Latin, 1451. Ptolemy, Public domain, via Wikimedia Commons

Science and philosophy in the world according to Ptolemy

The first two are devoted to the philosophical and scientific dimension of Ptolemaic theories, Ptolemy exposes a mathematical conception of the universe, as well as the bases of trigonometry.

The movement of the Sun

In the next book, Ptolemy is busy determining the motion of the Sun. He decides between two choices, an eccentric movement (trajectory whose center does not coincide with that of the Earth) and an epicyclic movement (the trajectory of the sun taking into account the deferent). He opted for the eccentric movement.

The movement of the moon

It is the Moon that is treated in books 4, 5, 6 and 7. To measure its movement, he was obliged to adopt a system of epicycles whose deferent is eccentric. This model is not suitable for quadratures (1st and last quarter of the Moon) but it still correctly describes eclipses.

Stellar Catalog

The eighth and ninth books catalog and reference the stars: 1022 celestial bodies of the Milky Way organized into 48 constellations. Measured with precision (within 1/6°), their magnitude is divided into six classes. This catalog dates from AD 137. J.C.

Physical and orbital characteristics of the planets

The following books deal with the planets, their dimensions, their retrograde movement for some which pose problems for the astronomer, the arcs, their planetary orbit…
He concludes his work on the problems of heliacal rising and setting of the planets (rising before the Sun or setting after the Sun).

Astronomical coordinate systems: horizontal, equatorial, ecliptic, galactic and supergalactic

Sources: PinterPandai, Britannica, University of St Andrews (Scotland), Famous Scientists

Photo credit (main picture): Jan van Loon, Public domain, via Wikimedia Commons

Photo description: Andreas Cellarius : Scenographia Systematis Mundani Ptolemaici. Excerpt from Harmonia Macroscomica by Andreas Cellarius, 1660/61. Figure showing the signs of the zodiac and the solar system with the Earth at its center, according to the Ptolemy system.

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